1. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Itay Hen Information Sciences Institute, USC NIPS Quantum Machine Learning Workshop December 12, 2015 Fidelity-Optimized Quantum State Estimation Joint work with Amir Kalev, UNM

2. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Disclaimer disclaimer: no Quantum Machine Learning per se here, Disclaimer: 1) a renunciation of any claim to or connection with; 2) disavowal; 3) a statement made to save one’s own ass. but… machine learning for quantum systems

3. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop The Problem an oven is emitting identical copies of the same unknown quantum state in a steady flow. objective: find out what this state is what measurements should we perform? oven measurement apparatus.........

4. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop The Problem what is the optimal sequence of measurements that would yield the best estimate with the smallest error and in the least amount of measurements? oven measurement apparatus......... an oven is emitting identical copies of the same unknown quantum state in a steady flow.

5. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  some probability theory  the protocol  some results  conclusions and applications for actual quantum machine learning Outline

6. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Probability theory

7. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  given an emitted state, what is the probability of getting the outcome ? Some probability theory

8. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  given an emitted state, what is the probability of getting the outcome after a single measurement?  what is the probability of getting the sequence of outcomes ? Some probability theory

9. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  now, given the sequence of outcomes, what is the probability that the emitted state is ? for that, we have Bayes’ law: Some probability theory  given an emitted state, what is the probability of getting the outcome after a single measurement?  what is the probability of getting the sequence of outcomes ?

10. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  now, given the sequence of outcomes, what is the probability that the emitted state is ? for that, we have Bayes’ law: Some probability theory probability of getting the state given the sequence of outcomes probability of getting the sequence of outcomes given the state the a priori probability of the state the probability of obtaining the sequence of outcomes  let us assume for simplicity (we don’t have to) that we have no knowledge about oven, i.e., that.  moreover, we have.

11. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  we thus end up with:  where: Some probability theory

12. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop The protocol

13. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  equipped with the above probability measure, we can give a general “learning” protocol for optimized adaptive tomography: The protocol 1. perform measurement in a randomly-chosen basis 2. based on record of measurement outcomes thus far, find most-likely state 4. compute optimal basis for next measurement 5. execute measurement in optimal basis 3. exit if convergence criterion has been reached, otherwise:  remaining questions: how do we calculate most-likely state / best guess? how do we determine the optimal measurement basis?

14. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  given a list of outcomes from all measurements thus far. what should be our best guess in the k-th step for the emitted state ?  this actually depends on how we define “best”. let’s say we’d like to maximize the fidelity of our guess with the real thing.  obviously, we don’t know what is, but we know the probability of occurrence for each state, so we can guess:  plugging in what we already have for, we get: Most likely state

15. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  now, we can rewrite  as:  put differently: Most likely state where and

16. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  given a list of prior measurement outcomes, how shall we determine the next basis of measurements?  is there a simple clear answer?  we have to carefully state what we would like accomplished. Determining next basis of measurement we would like to maximize the fidelity of the emitted state with our best guess after the measurement

17. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  how do we do that?  let’s say that the chosen basis of measurement in the k-th step is: Determining next basis of measurement  let’s assume that after the measurement is carried out, the obtained outcome is with.  we would like to maximize the fidelity of our best guess based on all outcomes “so far” with the real state. but, we have already calculated that, it’s simply

18. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  of course, we do not know which of the outcomes we’ll get. Determining next basis of measurement  we must therefore average over all possible outcomes, namely:  here, is the probability of obtaining the n-th outcome given what we know so far about the emitted state:

19. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  but what is: exactly? Determining next basis of measurement  it’s:  putting it all together, we find that the optimal basis is simply: where and

20. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Some Results

21. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  example: an oven is emitting copies of a qubit.  let’s say the following outcomes have been obtained: 4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.  in which direction should the next measurement be performed?  first, what’s the “most likely” state? Next basis of measurement: an example oven measurement apparatus.........

22. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  example: an oven is emitting copies of a qubit.  let’s say the following outcomes have been obtained: 4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.  in which direction should the next measurement be performed?  clearly, the best guess is up-z. Next basis of measurement: an example oven measurement apparatus.........

23. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Meaning of results  what does the requirement of maximizing where mean exactly?  it tells us that we should find a basis of measurements such that all outcomes are equally probable!  it tells us to perform a measurement in a basis that we cannot possibly guess what the outcome is!  putting it all together, we arrive at: and

24. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  going back to the example, outcomes are: 4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.  in which direction should the next measurement be performed?  if we perform a measurement in the z-direction, we have a pretty good guess of what of the outcome is going to be.  this is not the case in the x and y directions. but, there’s more certainty in the x direction. Meaning of results

25. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  going back to the example, outcomes are: 4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.  in which direction should the next measurement be performed?  if we perform a measurement in the z-direction, we have a pretty good guess of what of the outcome is going to be.  this is not the case in the x and y directions. but, there’s more certainty in the x direction.  we should therefore measure in the y direction! Meaning of results

26. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  consider the first few iterations of the protocol in the qubit case  an oven is emitting qubits one by one…  protocol dictates that we perform the first measurement in some random direction. let’s call the outcome up-z.  what does the protocol say about next basis of measurement? The qubit case: first few measurements

27. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  consider the first few iterations of the protocol in the qubit case  an oven is emitting qubits one by one…  protocol dictates that we perform the first measurement in some random direction. let’s call the outcome up-z.  what does the protocol say about next basis of measurement?  should be in a basis “orthogonal to z”. measurement direction should be on equator of Bloch sphere. let’s call the outcome up-x.  what about the next measurement basis? orthogonal to z and x, namely y.  next one is more complicated… The qubit case: first few measurements

28. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  repeatedly performing a numerical experiment thousands of times, we calculated the mean infidelity (with respect to the true state) as a function of number of measurements.  compared several methods: random-basis measurements. repeated measurements in the x, y, z directions. x, y, z measurements chosen in optimal way. fully-optimized.  no surprise, learning methods are superior. Numerical results: the qubit case

29. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  another example: qudit with d=4. again, mean infidelity as a function of number of measurements.  here, we’re assuming that the available measurements are only “local Pauli”, i.e., xx, xy, xz,…,zz.  comparing a random sequence of local Pauli measurements with an optimized sequence. Numerical results: the qudit case (d=4)

30. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Conclusions and what’s next

31. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop  optimized adaptive tomography helps!  easily extended to emitted mixed states, generalized measurements, etc.  what’s next? we have seen that the first few optimizations of the measurement bases yield “orthogonal” or, mutually unbiased, bases. this procedure can therefore be used to generate sets of MUBs (or, so we believe).  can be carried over to machine learning protocols, e.g., Wiebe et al’s “Quantum Hamiltonian Learning”. in some situations, learning curve can be optimized, provided that we can utilize all the gathered information to maximize our knowledge about desired quantities. Conclusions

32. Itay Hen Dec 12, 2015 NIPS Quantum Machine Learning Workshop Itay Hen Information Sciences Institute, USC NIPS Quantum Machine Learning Workshop December 12, 2015 Fidelity Optimized Quantum State Estimation Joint work with Amir Kalev, UNM Thank You!